Rotate $y=a(\cos(x)+b)$ by $2\pi$ radians about the $x$-axis between $x=c\pi$ and $x=(c+2)\pi$. Find the volume of revolution.

Rotate the graph of  $y=a\ln(bx)$  by $2\pi$ radians about the $y$-axis between $y=c$ and $y=d$. Find the volume of revolution.

Calculate the Pearson correlation coefficient on paired data and comment on the significance.

Calculate the Pearson correlation coefficient on paired data and comment on the significance.

Spearman rank correlation calculated. 8 paired observations.

Spearman rank correlation calculated. 10 paired observations.

Elementary examples of multiplication and powers of complex numbers. Four parts.

Inverse and division of complex numbers.  Four parts.

Multiplication of complex numbers. Four parts.

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$. 

Hence, or otherwise, find roots of  $ax^2+bx+c=0$.

Includes a video which shows how to solve a quadratic by completing the square.

Express $\displaystyle a \pm  \frac{c}{x + d}$ as an algebraic single fraction.

Express $\displaystyle b+  \frac{dx+p}{x + q}$ as an algebraic single fraction. 

Express $\displaystyle ax+b+  \frac{dx+p}{x + q}$ as an algebraic single fraction. 

Express $\displaystyle \frac{ax+b}{x + c} \pm  \frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator. 

Express $\displaystyle \frac{ax+b}{cx + d} \pm  \frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator. 

First part: Express  $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d},\;a=-c$. Numerator is an integer.

Second part: $\displaystyle \frac{a}{px + b} +\frac{c}{qx + d}+ \frac{r}{sx+t}$ as single fraction 

First part: express as a single fraction: $\displaystyle \frac{a}{x + b} +  \frac{c}{x + d},\; a \neq -c$.

Second part: Find $\displaystyle \frac{a}{x + b} + \frac{c}{x + d}+\frac{r}{x+t}$ as a single fraction.

First part: express as a single fraction: $\displaystyle \frac{a}{px + b} +  \frac{c}{qx + d}$.

Second part: Find $\displaystyle \frac{a}{px + b} + \frac{c}{qx + d}+\frac{r}{sx+t}$ as a single fraction.

Express $\displaystyle \frac{a}{x + b} + \frac{cx+d}{x^2 +px+ q}$ as an algebraic single fraction over a common denominator. 

Express $\displaystyle \frac{a}{x + b} +\frac{c}{(x + b)^2}$ as an algebraic single fraction.

Express $\displaystyle \frac{a}{x + b} +\frac{cx+d}{(x + b)^2}$ as an algebraic single fraction.

Express $\displaystyle \frac{a}{(x+r)(px + b)} + \frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$.

The derivative of $\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$. Find $g(x)$.

Differentiate $\displaystyle \ln((ax+b)^{m})$

Differentiate

\[ \sqrt{a x^m+b})\]

Differentiate $\displaystyle e^{ax^{m} +bx^2+c}$

Differentiate the following functions: $\displaystyle x ^ n \sinh(ax + b),\;\tanh(cx+d),\;\ln(\cosh(px+q))$

Differentiate $\displaystyle \cos(e^{ax}+bx^2+c)$.

Contains a video solving a similar example.

Express $\displaystyle \frac{a}{x + b} \pm  \frac{c}{x + d}$ as an algebraic single fraction over a common denominator. 

Contains a video in Show steps.

This is an old demo. For a demonstration of Numbas's current features, please see the newest demo exam.

A few questions which show off various features of Numbas.